The simplest form of `0.12bar3` is

Options

A. `(41)/(330)`
B. `(37)/(333)`
C. `(41)/(333)`
D. None of these

Answer

To find the simplest form of \(0.12\overline{3}\), we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 0.12\overline{3} \). This means \( x = 0.1233333...\), where the digit 3 repeats indefinitely.

### Step 2: Multiply by 10
To eliminate the repeating part, we can multiply both sides of the equation by 10:
\[ 10x = 1.233333...\, \]
This shifts the decimal point one place to the right.

### Step 3: Multiply by 100
Next, we multiply both sides of the original equation by 100 to align the repeating parts:
\[ 100x = 12.33333...\, \]

### Step 4: Set up the equations
Now we have two equations:
1. \( 10x = 1.23333...\)
2. \( 100x = 12.33333...\)

### Step 5: Subtract the first equation from the second
We will subtract the first equation from the second to eliminate the repeating part:
\[ 100x - 10x = 12.33333... - 1.23333... \]
This simplifies to:
\[ 90x = 11.1 \]

### Step 6: Simplify the right side
To simplify \( 11.1 \), we can express it as a fraction:
\[ 11.1 = \frac{111}{10} \]
So, we have:
\[ 90x = \frac{111}{10} \]

### Step 7: Solve for \( x \)
Now we can solve for \( x \):
\[ x = \frac{111}{10 \times 90} = \frac{111}{900} \]

### Step 8: Simplify the fraction
Next, we simplify \( \frac{111}{900} \). Both the numerator and denominator can be divided by 3:
\[ \frac{111 \div 3}{900 \div 3} = \frac{37}{300} \]

### Conclusion
Thus, the simplest form of \( 0.12\overline{3} \) is:
\[ \frac{37}{300} \]